Metamath Proof Explorer


Theorem cdleme32e

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 20-Feb-2013)

Ref Expression
Hypotheses cdleme32.b 𝐵 = ( Base ‘ 𝐾 )
cdleme32.l = ( le ‘ 𝐾 )
cdleme32.j = ( join ‘ 𝐾 )
cdleme32.m = ( meet ‘ 𝐾 )
cdleme32.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme32.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme32.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme32.c 𝐶 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
cdleme32.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdleme32.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
cdleme32.i 𝐼 = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) )
cdleme32.n 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐶 )
cdleme32.o 𝑂 = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ( 𝑥 𝑊 ) ) ) )
cdleme32.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , 𝑂 , 𝑥 ) )
Assertion cdleme32e ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝐹𝑋 ) ( 𝐹𝑌 ) )

Proof

Step Hyp Ref Expression
1 cdleme32.b 𝐵 = ( Base ‘ 𝐾 )
2 cdleme32.l = ( le ‘ 𝐾 )
3 cdleme32.j = ( join ‘ 𝐾 )
4 cdleme32.m = ( meet ‘ 𝐾 )
5 cdleme32.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme32.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme32.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdleme32.c 𝐶 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
9 cdleme32.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
10 cdleme32.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
11 cdleme32.i 𝐼 = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) )
12 cdleme32.n 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐶 )
13 cdleme32.o 𝑂 = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ( 𝑥 𝑊 ) ) ) )
14 cdleme32.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , 𝑂 , 𝑥 ) )
15 simp23l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → 𝑃𝑄 )
16 15 pm2.24d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( ¬ 𝑃𝑄𝑋 ( 𝑁 ( 𝑌 𝑊 ) ) ) )
17 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → 𝐾 ∈ HL )
18 17 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → 𝐾 ∈ Lat )
19 simp21l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → 𝑋𝐵 )
20 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → 𝑊𝐻 )
21 1 6 lhpbase ( 𝑊𝐻𝑊𝐵 )
22 20 21 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → 𝑊𝐵 )
23 1 2 4 latleeqm1 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵 ) → ( 𝑋 𝑊 ↔ ( 𝑋 𝑊 ) = 𝑋 ) )
24 18 19 22 23 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑋 𝑊 ↔ ( 𝑋 𝑊 ) = 𝑋 ) )
25 1 4 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵 ) → ( 𝑋 𝑊 ) ∈ 𝐵 )
26 18 19 22 25 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑋 𝑊 ) ∈ 𝐵 )
27 simp21r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → 𝑌𝐵 )
28 1 4 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵 ) → ( 𝑌 𝑊 ) ∈ 𝐵 )
29 18 27 22 28 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑌 𝑊 ) ∈ 𝐵 )
30 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
31 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
32 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
33 simp31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 cdleme27cl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) → 𝑁𝐵 )
35 30 31 32 33 15 34 syl122anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → 𝑁𝐵 )
36 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑁𝐵 ∧ ( 𝑌 𝑊 ) ∈ 𝐵 ) → ( 𝑁 ( 𝑌 𝑊 ) ) ∈ 𝐵 )
37 18 35 29 36 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑁 ( 𝑌 𝑊 ) ) ∈ 𝐵 )
38 simp33 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → 𝑋 𝑌 )
39 1 2 4 latmlem1 ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵𝑊𝐵 ) ) → ( 𝑋 𝑌 → ( 𝑋 𝑊 ) ( 𝑌 𝑊 ) ) )
40 18 19 27 22 39 syl13anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑋 𝑌 → ( 𝑋 𝑊 ) ( 𝑌 𝑊 ) ) )
41 38 40 mpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑋 𝑊 ) ( 𝑌 𝑊 ) )
42 1 2 3 latlej2 ( ( 𝐾 ∈ Lat ∧ 𝑁𝐵 ∧ ( 𝑌 𝑊 ) ∈ 𝐵 ) → ( 𝑌 𝑊 ) ( 𝑁 ( 𝑌 𝑊 ) ) )
43 18 35 29 42 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑌 𝑊 ) ( 𝑁 ( 𝑌 𝑊 ) ) )
44 1 2 18 26 29 37 41 43 lattrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑋 𝑊 ) ( 𝑁 ( 𝑌 𝑊 ) ) )
45 breq1 ( ( 𝑋 𝑊 ) = 𝑋 → ( ( 𝑋 𝑊 ) ( 𝑁 ( 𝑌 𝑊 ) ) ↔ 𝑋 ( 𝑁 ( 𝑌 𝑊 ) ) ) )
46 44 45 syl5ibcom ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( ( 𝑋 𝑊 ) = 𝑋𝑋 ( 𝑁 ( 𝑌 𝑊 ) ) ) )
47 24 46 sylbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑋 𝑊𝑋 ( 𝑁 ( 𝑌 𝑊 ) ) ) )
48 simp22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) )
49 pm4.53 ( ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ↔ ( ¬ 𝑃𝑄𝑋 𝑊 ) )
50 48 49 sylib ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( ¬ 𝑃𝑄𝑋 𝑊 ) )
51 16 47 50 mpjaod ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → 𝑋 ( 𝑁 ( 𝑌 𝑊 ) ) )
52 14 cdleme31fv2 ( ( 𝑋𝐵 ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ) → ( 𝐹𝑋 ) = 𝑋 )
53 19 48 52 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝐹𝑋 ) = 𝑋 )
54 simp1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) )
55 simp23 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) )
56 simp32 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 )
57 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme32a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑌𝐵 ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝐹𝑌 ) = ( 𝑁 ( 𝑌 𝑊 ) ) )
58 54 27 55 33 56 57 syl122anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝐹𝑌 ) = ( 𝑁 ( 𝑌 𝑊 ) ) )
59 51 53 58 3brtr4d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑋𝐵𝑌𝐵 ) ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑋 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑠 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) → ( 𝐹𝑋 ) ( 𝐹𝑌 ) )